| ■No23119に返信(ミニーさんの記事) > 簡単にいきました 解決済み! でしょうが
Qの拡大体Q(Sqrt[3]+Sqrt[5])達の視座から ; In[2]:= f1 = x1^2 - 3; f2 = x2^2 - 5; f = x - (x1 + x2); GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}] Solve[% == {0, 0, 0}]; GB[[1]]
Out[5]= {4 - 16*x^2 + x^4, 18*x - x^3 - 4*x2, 14*x - x^3 + 4*x1}
Out[7]= 4 - 16*x^2 + x^4
In[8]:= f = GB[[1]]; g = x; {f, g} {d, {X0, Y0}} = PolynomialExtendedGCD[f, g] {{f*X0}, {g*Y0}}
Out[9]= {4 - 16*x^2 + x^4, x}
Out[10]= {1, {1/4, 4*x - x^3/4}}
Out[11]= {{1/4*(4 - 16*x^2 + x^4)}, {x*(4*x - x^3/4)}}
In[12]:= Y0 /. x -> Sqrt[3] + Sqrt[5] Expand[%] A1 = Expand[%*2] N[%]
Out[12]= 4*(Sqrt[3] + Sqrt[5]) - 1/4*(Sqrt[3] + Sqrt[5])^3
Out[13]= -(Sqrt[3]/2) + Sqrt[5]/2
Out[14]= -Sqrt[3] + Sqrt[5]
Out[15]= 0.5040171699309126
In[16]:= f1 = x1^2 - 5; f2 = x2^2 - 7; f = x - (x1 + x2); GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}] Solve[% == {0, 0, 0}]; GB[[1]]
Out[19]= {4 - 24*x^2 + x^4, 26*x - x^3 - 4*x2, 22*x - x^3 + 4*x1}
Out[21]= 4 - 24*x^2 + x^4
In[22]:= f = GB[[1]]; g = x; {f, g} {d, {X0, Y0}} = PolynomialExtendedGCD[f, g] {{f*X0}, {g*Y0}}
Out[23]= {4 - 24*x^2 + x^4, x}
Out[24]= {1, {1/4, 6*x - x^3/4}}
Out[25]= {{1/4*(4 - 24*x^2 + x^4)}, {x*(6*x - x^3/4)}}
In[26]:= Y0 /. x -> Sqrt[5] + Sqrt[7] Expand[%] A2 = Expand[%*2] N[%]
Out[26]= 6*(Sqrt[5] + Sqrt[7]) - 1/4*(Sqrt[5] + Sqrt[7])^3
Out[27]= -(Sqrt[5]/2) + Sqrt[7]/2
Out[28]= -Sqrt[5] + Sqrt[7]
Out[29]= 0.4096833335648009
In[30]:= f1 = x1^2 - 7; f2 = x2^2 - 9; f = x - (x1 + x2); GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}] Solve[% == {0, 0, 0}]; GB[[1]]
Out[33]= {4 - 32*x^2 + x^4, 34*x - x^3 - 4*x2, 30*x - x^3 + 4*x1}
Out[35]= 4 - 32*x^2 + x^4
In[36]:= f = GB[[1]]; g = x; {f, g} {d, {X0, Y0}} = PolynomialExtendedGCD[f, g] {{f*X0}, {g*Y0}}
Out[37]= {4 - 32*x^2 + x^4, x}
Out[38]= {1, {1/4, 8*x - x^3/4}}
Out[39]= {{1/4*(4 - 32*x^2 + x^4)}, {x*(8*x - x^3/4)}}
In[40]:= Y0 /. x -> Sqrt[7] + Sqrt[9] Expand[%] A3 = Expand[%*2] N[%]
Out[40]= 8*(3 + Sqrt[7]) - 1/4*(3 + Sqrt[7])^3
Out[41]= 3/2 - Sqrt[7]/2
Out[42]= 3 - Sqrt[7]
Out[43]= 0.3542486889354093
In[44]:= f1 = x1^2 - 9; f2 = x2^2 - 11; f = x - (x1 + x2); GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}] Solve[% == {0, 0, 0}]; GB[[1]]
Out[47]= {4 - 40*x^2 + x^4, 42*x - x^3 - 4*x2, 38*x - x^3 + 4*x1}
Out[49]= 4 - 40*x^2 + x^4
In[50]:= f = GB[[1]]; g = x; {f, g} {d, {X0, Y0}} = PolynomialExtendedGCD[f, g] {{f*X0}, {g*Y0}}
Out[51]= {4 - 40*x^2 + x^4, x}
Out[52]= {1, {1/4, 10*x - x^3/4}}
Out[53]= {{1/4*(4 - 40*x^2 + x^4)}, {x*(10*x - x^3/4)}}
In[54]:= Y0 /. x -> Sqrt[9] + Sqrt[11] Expand[%] A4 = Expand[%*2] N[%]
Out[54]= 10*(3 + Sqrt[11]) - 1/4*(3 + Sqrt[11])^3
Out[55]= -(3/2) + Sqrt[11]/2
Out[56]= -3 + Sqrt[11]
Out[57]= 0.3166247903553998
In[58]:= {A1, A2, A3, A4}
Out[58]= {-Sqrt[3] + Sqrt[5], -Sqrt[5] + Sqrt[7], 3 - Sqrt[7], -3 + Sqrt[11]} In[59]:= A1 + A2 + A3 + A4 N[%] Out[59]= -Sqrt[3] + Sqrt[11]<--コタエ Out[60]= 1.5845739827865226
時には ▼分母に無理数がない▼ 子のように 誰にもI(deal)を話せない... なんて 商環 k[X]/I I(deal)は地球を救う とか 巷で 手ごろのが在るので; http://blog.livedoor.jp/seven_triton/archives/2006-05.html 様 の √素数の問題 の 頁 に; 問4をどうぞ。 -------------------------------------------------
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